Linear precoding design for massive MIMO based on the minimum mean square error algorithm
 Zhou Ge^{1}Email author and
 Wu Haiyan^{2}
DOI: 10.1186/s1363901600644
© The Author(s). 2017
Received: 10 March 2016
Accepted: 29 November 2016
Published: 25 January 2017
Abstract
Compared with the traditional multipleinput multipleoutput (MIMO) systems, the large number of the transmit antennas of massive MIMO makes it more dependent on the limited feedback in practical systems. In this paper, we study the problem of precoding design for a massive MIMO system with limited feedback via minimizing mean square error (MSE). The feedback from mobile users to the base station (BS) is firstly considered; the BS can obtain the quantized information regarding the direction of the channels. Then, the precoding is designed by considering the effect of both noise term and quantization error under transmit power constraint. Simulation results show that the proposed scheme is robust to the channel uncertainties caused by quantization errors.
Keywords
Massive MIMO Precoding Minimum mean square error (MMSE) Limited feedback1 Introduction
Multipleinput multipleoutput (MIMO) techniques have gained considerable attention in modern wireless communications since it can significantly improve the capacity and reliability of wireless systems [1]. The essence of downlink multiuser MIMO is precoding, which means that the antenna arrays are used to direct each data signal spatially towards its intended receiver. Unfortunately, the precoding design in multiuser MIMO requires very accurate instantaneous channel state information (CSI) [2] which can be cumbersome to achieve in practice. To further achieve more dramatic gains as well as to simplify the required signal processing, massive MIMO techniques have been proposed in [3, 4] by installing a large number of antennas at base stations (BS), possibly in the order of tens or hundreds, which promises significant performance gains in terms of spectral efficiency and energy efficiency compared with conventional MIMO and is becoming a cornerstone of future 5G systems [5, 6].
From a practical point of view, realizing massive MIMO systems has to deal with several challenges, one of which is the lowcomplexity and nearoptimal precoding scheme [7, 8]. Generally, precoding approaches can be classified into nonlinear precoding and linear precoding. The optimal precoding is the nonlinear dirty paper precoding (DPC) [9], which can effectively eliminate the interference between different users and achieve optimal performance. However, nonlinear precoding schemes usually suffer from high complexity which makes them unpractical due to the hundreds of antennas in massive MIMO systems. Since the asymptotic orthogonality of massive MIMO channel matrix, simple linear precoding (e.g., zeroforcing (ZF) precoding) can be used to achieve capacityapproaching performance. Nevertheless, ZF precoding requires matrix inversion of very large size, which exhibits prohibitively high complexity. To reduce the complexity of matrix inversion of large size, a Neumannbased precoding is proposed in [10] to reduce the computational complexity in an iterative method, but the required complexity is still unaffordable. Recently, a low peaktoaverage power ratio (PAPR) precoding based on the approximate message passing (AMP) algorithm [11] and a successive overrelaxation (SOR)based precoding [12] are respectively proposed to minimize multiuser interference (MUI) in massive multiuser MIMO systems. The aforementioned works are based on the assumption of a perfect CSI at the BS, which is somewhat too optimistic for practical applications. As a result, it is essential to investigate the robust precoding design in massive MIMO systems.
Inspired by the abovementioned works, in this paper, we study the precoding design for a singlecell downlink massive MIMO system with limited feedback, while guaranteeing transmit power constraint. Each user terminal (UT) feeds back the quantized side information to BS to assist its transmission. We propose a linear precoder design scheme via the minimum mean square error (MMSE) criteria with respect to the CSI imperfection. The proposed scheme is an improved approach, which is robust to the channel uncertainties caused by quantization errors and the lack of channel quality information (CQI).
The rest of this paper is organized as follows. In Section 1, the system mode of massive MIMO is introduced and the problem is formulated. In Section 2, a linear precoder based on MMSE criteria is designed by considering the impact of the noise term and CSI quantization error. Numerical results are presented in Section 3. Finally, concluding remarks are made in Section 4.
Notations: Throughout this paper, boldface lowercase and uppercase letters denote vectors and matrices, respectively. The transpose, conjugate transpose, trace, and Frobenius norm of a matrix A are denoted as A ^{T}, A ^{H}, tr(A), ‖A‖_{ F }, respectively. I _{ M × M } denotes a M × M identity matrix. Ε[⋅] denotes the expectation operator. diag(⋅) stands for a diagonal matrix with the given elements on the diagonal. Re(⋅) represents the real part of the input.
2 System model
We assume that each UT can perfectly estimate the downlink CSI and send it back to ST using local feedback. All of the feedback channels are assumed to be noiseless and delay free. To facilitate analysis, the channel is decomposed into the channel direction information (CDI) and CQI [11]. The kth UT estimates the CSI of channel D _{ k } perfectly and quantizes the CDI \( {\tilde{\mathbf{D}}}_k={\mathbf{D}}_k/\left\Vert {\mathbf{D}}_k\right\Vert \) to a unit norm vector \( {\widehat{\mathbf{D}}}_k. \)
3 Linear precoder design
In this section, we introduce a linear precoder design scheme that considers the effect of both noise term and quantization error.
3.1 Channel model
3.2 Linear precoder design
where \( \alpha {\mathbf{I}}_{N_R}= E\left[\mathbf{A}\right]= E\left[{\left({\mathbf{I}}_{N_R}{\mathbf{Z}}^{\mathrm{H}}\mathbf{Z}\right)}^{1/2}\mathbf{R}\right]. \)
where \( \rho =\sqrt{\frac{P_T}{\rho {\left(\widehat{\mathbf{D}}{\widehat{\mathbf{D}}}^{\mathrm{H}}+\eta {\mathbf{I}}_{N_R}\right)}^{1}\widehat{\mathbf{D}}}} \) (ρ is normalized power factor) and \( \eta =\frac{P_T{N}_T\varDelta +{N}_R\left({N}_T{n}_r\right)}{P_T\left[\left({N}_T{n}_r\right){N}_T^2\varDelta \right]}. \)

(1) If Δ = 0, that is perfect CSI. Here, \( \eta =\frac{N_R}{P_T} \). The precoding matrix can be rewritten as

$$ \mathbf{W}=\rho {\left(\widehat{\mathbf{D}}{\widehat{\mathbf{D}}}^{\mathrm{H}}+\eta {\mathbf{I}}_{N_R}\right)}^{1}\widehat{\mathbf{D}} $$(16)

(2) If Δ ≠ 0, that is imperfect CSI. Here, n _{ r } = 1 and N _{ T } = N _{ R } = K for simplicity, \( \varDelta =\frac{N_T1}{N_T}{2}^{\frac{B}{N_T1}}. \). The precoding matrix can be rewritten as

$$ \mathbf{W}=\rho {\left[\widehat{\mathbf{D}}{\widehat{\mathbf{D}}}^{\mathrm{H}}+\frac{P_T{2}^{\frac{B}{N_T1}}+{N}_T}{P_T\left(1{N}_T{2}^{\frac{B}{N_T1}}\right)}{\mathbf{I}}_{N_R}\right]}^{1}\widehat{\mathbf{D}}. $$(17)
Here, we use the subgradient algorithm to solve the problem. Using a constant step length t _{1} and t _{2}, the subgradient algorithm can converge to the optimal point of convex problems within a small range.
Since MMSE function in (12) is convex on a single precoder W, updating W at each iteration monotonically reduces the MMSE in (12), which is lower bounded by zero. Algorithm 1 can converge to the optimal point of the problem (6) within a small range. Although the precoder design depends on inaccurate CSI feedback, it may not always satisfy the transmit power constraint. However, we can assume a procedure that a feedforward link exists between UTs and BS. Each UT sends information from the precoder W to the BS via the feedforward link, then BS estimates the received power to satisfy the transmit power constraint.
3.3 Analysis of computational complexity
4 Simulation results
In this section, the performance of the proposed scheme is evaluated by a computer simulation. In our simulations, the elements of all the signaling channel matrices are assumed to be i.i.d. complex Gaussian variables with zero means and unit variance. We assume that the number of UTs is K = 30, the total transmit power is P _{ T } = 20 dB, and the background noise is σ ^{2} = 1.
For simplicity, we assume that the received number of antennas at each UT is n _{ r } = 1.
5 Conclusions
In this paper, we investigated the problem of linear precoding design for massive MIMO system in a single cell based on MMSE criteria under transmit power constraint. The proposed scheme was robust to the uncertainties in the CSI as it taken into account the effect of quantization errors and noise term. Simulation results show the superiority of our proposed quantization scheme. In the future work, we plan to study the partial feedback of CSI for multicell massive MIMO systems.
Declarations
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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